Solving Equations

Hi everyone,

I was working on some solving equations in preparation for university, and came across a few that I just could not understand how to do them. I have the answers, but do not know how to get to them, how painful....

1. Find a cubic function of x whose graph has x-intercepts equal to -1 and 2, and in which f(0)=6 and f(3)=12.

Answer: 2x^3-5x^2-x+6

2. The roots of the equation x^2+px+1=0 are a and b, and the roots of the equation x^2-9x+q=0 are a+2b and 2a+b. (a and b are greek symbols).

Answer: p=-3, q=19

3. If the roots of the equation x^3-15x^2+cx-105, are a, a+b, and a-b, determine the values of a,b, and c.

I was working on some solving equations in preparation for university, and came across a few that I just could not understand how to do them. I have the answers, but do not know how to get to them, how painful....

1. Find a cubic function of x whose graph has x-intercepts equal to -1 and 2, and in which f(0)=6 and f(3)=12.

Answer: 2x^3-5x^2-x+6

[snip]

Use the model (why?) and use the fact that f(0) = 6 and f(3) = 12 to get two simuatenous equations for a and b.

Originally Posted by KelvinScale

[snip]
2. The roots of the equation x^2+px+1=0 are a and b, and the roots of the equation x^2-9x+q=0 are a+2b and 2a+b. (a and b are greek symbols).

Answer: p=-3, q=19

[snip]

. Therefore:

.... (1)

.... (2)

In a similar way, . Therefore:

...... (3)

....... (4)

Solve equations (1), (2), (3) and (4) simultaneously to get p and q.

If you need more help, please show all your work and say where you get stuck.